There are positive integers that have these properties:

$\bullet$ I. The sum of the squares of their digits is $50,$ and

$\bullet$ II. Each digit is larger than the one on its left.

What is the product of the digits of the largest integer with both properties?
Solution: To meet the first condition, numbers which sum to $50$ must be chosen from the set of squares $\{1, 4, 9, 16, 25, 36, 49\}.$ To meet the second condition, the squares selected must be different. Consequently, there are three possibilities: $1+49,$ $1+4+9+36,$ and $9+16+25.$ These correspond to the integers $17,$ $1236,$ and $345,$ respectively. The largest is $1236,$ and the product of its digits is $1\cdot2\cdot3\cdot6=\boxed{36}.$